As a preparation to investigate the possibility to explain the 12.5-yr NANOGrav result, we here consider the low-scale phase transition.
Four crucial parameters for the estimation of the GW from FOPT are: phase transition temperature \(T_\star \) , phase transition strength \(\alpha \) , the inverse duration of the phase transition, and the bubble wall velocity. We take \(\alpha =\frac{\Delta \rho }{\rho _R}\;\)  with the \(\rho _R\)  and \(\Delta \rho \)  being the radiative energy in the plasma and the released latent heat. The inverse time duration of the phase transition is obtained as \(\frac{\beta }{H_n}=T\frac{d (S_3(T)/T)}{d T}|_{T=T_n}\; \) .
For magnification of the GW, we take the wall velocity \(v_b\approx 1\)  with the calculation method follows Ref. [1]}. In addition, for a large \(\alpha \) , the duration of the phase transition would be very important [2]}, [3]}, [4]}, [5]}, [6]}. For more details on phase transition and gravitational wave calculations, see Appendix.
